Implications

This non-implication, Form 404 \( \not \Rightarrow \) Form 133, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10048, whose string of implications is:
    9 \(\Rightarrow\) 404
  • A proven non-implication whose code is 3. In this case, it's Code 3: 174, Form 9 \( \not \Rightarrow \) Form 341 whose summary information is:
    Hypothesis Statement
    Form 9 <p>Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) <a href="/books/8">Jech [1973b]</a>: \(E(I,IV)\) <a href="/articles/Howard-Yorke-1989">Howard/Yorke [1989]</a>): Every Dedekind finite set is finite. </p>

    Conclusion Statement
    Form 341 <p> Every Lindel&ouml;f metric space is second countable. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7599, whose string of implications is:
    133 \(\Rightarrow\) 340 \(\Rightarrow\) 341

The conclusion Form 404 \( \not \Rightarrow \) Form 133 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N58\) Keremedis/Tachtsis Model 2: For each \(n\in\omega-\{0\}\), let\(A_n=\{({i\over n}) (\cos t,\sin t): t\in [0.2\pi)\}\) and let the set of atoms\(A=\bigcup \{A_n: n\in\omega-\{0\}\}\) \(\cal G\) is the group of allpermutations on \(A\) which rotate the \(A_n\)'s by an angle \(\theta_n\), andsupports are finite

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